Question

Country k decides to reduce their $11.5 billion debt by a decay factor of 0.978 each year. this can be modeled by the equation a=11.5(0.978)ᵗ. estimate the debt after 13 years. round to the nearest hundredth and do not round until the final calculation.

Answer 1

To estimate the debt of Country K after 13 years, we use the exponential decay formula A = 11.5(0.978)^t with t=13. The calculation yields approximately $9.61 billion as the estimated debt.

Step-by-step explanation:

The student's question involves estimating the debt of Country K after 13 years given that it reduces by a decay factor of 0.978 each year. This can be calculated using the provided exponential decay formula A = 11.5(0.978)t, where A is the amount of debt after t years. To find the debt after 13 years, substitute t with 13 and calculate.

Here's the step-by-step calculation:

1. Write down the formula with the given values: A = 11.5(0.978)13.
2. Calculate the decay factor raised to the power of 13: (0.978)13.
3. Multiply the initial debt by the decay factor: 11.5 * (0.978)13.
4. Round the final result to the nearest hundredth and avoid rounding until the final calculation.

Performing the calculation, we get:

• (0.978)13 = 0.8352252258 (rounded to 10 decimal places)
• A = 11.5 billion * 0.8352252258 = 9.6050925977 billion
• Round to the nearest hundredth: A ≈ $9.61 billion